Fourier Analysis on Polytopes and the Geometry of Numbers (Student Mathematical Library)

This book offers a gentle introduction to the geometry of numbers from a modern Fourier-analytic point of view. One of the main themes is the transfer of geometric knowledge of a polytope to analytic knowledge of its Fourier transform. The Fourier transform preserves all of the information of a polytope, and turns its geometry into analysis. The approach is unique, and streamlines this emerging field by presenting new simple proofs of some basic results of the field. In addition, each chapter is fitted with many exercises, some of which have solutions and hints in an appendix. Thus, an individual learner will have an easier time absorbing the material on their own, or as part of a class. Overall, this book provides an introduction appropriate for an advanced undergraduate, a beginning graduate student, or researcher interested in exploring this important expanding field. Copyright Contents Acknowledgments Preface Introduction 1. Initial ideas 2. The Poisson summation formula Chapter 1. Motivational problem: Tiling a rectangle with rectangles 1. Intuition 2. Nice rectangles 3. Conventions and some definitions Notes Exercises Chapter 2. Examples nourish the theory 1. Intuition 2. Dimension 1—the classical sinc function 3. The Fourier transform of P as a complete invariant 4. Bernoulli polynomials 5. The cube and its Fourier transform 6. The standard simplex and its Fourier transform 7. Convex sets and polytopes 8. Any triangle and its Fourier transform 9. Stretching and translating 10. The parallelepiped and its Fourier transform 11. The cross-polytope 12. Observations and questions Notes Exercises Chapter 3. The basics of Fourier analysis 1. Intuition 2. Introducing the Fourier transform on L1(R^{d}) 3. The triangle inequality for integrals 4. The Riemann–Lebesgue lemma 5. The inverse Fourier transform 6. The torus R^{d}/Z^{d} 7. Piecewise smooth functions have convergent Fourier series 8. As f gets smoother, f̂ decays faster 9. How fast do Fourier coefficients decay? 10. The Schwartz space 11. Poisson summation I 12. Useful convergence lemmas in preparation for Poisson summation II 13. Poisson summation II: Á la Poisson 14. An initial taste of general lattices in anticipation of Chapter 5 15. Poisson summation III: For general lattices 16. The convolution operation 17. More relations between L1(R^{d}) and L2(R^{d}) 18. The Dirichlet kernel 19. The extension of the Fourier transform to L2: Plancherel 20. Approximate identities 21. Poisson summation IV: A practical Poisson summation formula 22. The Fourier transform of the ball 23. Uncertainty principles Notes Exercises Chapter 4. Geometry of numbers Part I: Minkowski meets Siegel 1. Intuition 2. Minkowski’s first convex body theorem 3. A Fourier transform identity for convex bodies 4. Tiling and multi-tiling Euclidean space by translations of polytopes 5. Extremal bodies 6. Zonotopes and centrally symmetric polytopes 7. Sums of two squares via Minkowski’s theorem 8. The volume of the ball and the sphere 9. Classical geometric inequalities 10. Minkowski’s theorems on linear forms 11. Poisson summation as the trace of a compact linear operator Notes Exercises Chapter 5. An introduction to Euclidean lattices 1. Intuition 2. Introduction to lattices 3. Sublattices 4. Discrete subgroups: An alternate definition of a lattice 5. Lattices defined by congruences 6. The Gram matrix 7. Dual lattices 8. Some important lattices 9. The Hermite normal form 10. The Voronoi cell of a lattice 11. Characters of lattices Notes Exercises Chapter 6. Geometry of numbers Part II: Blichfeldt’s theorems 1. Intuition 2. Blichfeldt’s theorem 3. Van der Corput’s inequality for convex bodies Notes Exercises Chapter 7. The Fourier transform of a polytope via its vertex description: Brion’s theorem 1. Intuition 2. Cones, simple polytopes, and simplicial polytopes 3. Tangent cones 4. The Brianchon–Gram identity 5. Brion’s formula for the Fourier transform of a simple polytope 6. Proof of Theorem 7.12, the Fourier transform of a simple polytope 7. The Fourier transform of any real polytope 8. Fourier–Laplace transforms of cones 9. The Fourier transform of a polygon 10. Each polytope has its moments 11. The zero set of the Fourier transform Notes Exercises Chapter 8. What is an angle in higher dimensions? 1. Intuition 2. Defining an angle in higher dimensions 3. Local solid angles for a polytope and Gaussian smoothing 4. 1-dimensional polytopes: Their angle polynomial 5. Pick’s formula and Nosarzewska’s inequality 6. The Gram relations for solid angles 7. Bounds for solid angles 8. The classical Euler–Maclaurin summation formula 9. Further topics Notes Exercises Appendix A. Solutions and hints to selected problems Solutions to Chapter 1 Solutions to Chapter 2 Solutions to Chapter 3 Solutions to Chapter 4 Solutions to Chapter 5 Solutions to Chapter 6 Solutions to Chapter 7 Appendix B. The dominated convergence theorem and other goodies 1. The dominated convergence theorem 2. Big-O and little-o 3. Various forms of convergence Credits for photographs and pictures Bibliography Index